# Problem Restatement A surveyor uses a chain to measure a distance of \( 300 \) m. However, the chain is \( .1 \) m **too long per 30 m**. What is the **actual distance**? --- # Step 1: Understand the Error - The chain is **.1 m too long** for every **30 m** measured. - That means, when the surveyor thinks they've measured 30 m, they've actually measured **less** than 30 m, because the chain is longer than it should be. ## For every 30 m (measured): - **Actual distance** covered = **Nominal length** × (True length / Nominal length) - But here, the **chain's true length = nominal length + .1 m** - So, for each 30 m measured, the **actual distance** is: \[ \text{Actual distance per 30 m} = 30 \times \frac{30}{30.1} \] --- # Step 2: Calculate Correction per 30 m \[ \text{Actual distance per 30 m} = 30 \times \frac{30}{30.1} = \frac{900}{30.1} \] \[ \frac{900}{30.1} \approx 29.9003 \text{ m} \] So, **when the surveyor thinks they have measured 30 m, they've actually measured only about 29.9003 m.** --- # Step 3: Find Number of Chain Lengths Used \[ \text{Number of 30 m lengths in 300 m} = \frac{300}{30} = 10 \] --- # Step 4: Compute Actual Distance The **actual total distance** is: \[ \text{Actual distance} = \text{Number of chains} \times \text{Actual per chain} \] \[ = 10 \times 29.9003 = 299.003 \text{ m} \] --- # Step 5: Alternatively, Use Correction Formula The **correction for incorrect chain length** is: \[ \text{Corrected distance} = \text{Measured distance} \times \frac{\text{True length of chain}}{\text{Nominal length of chain}} \] Here, - **Measured distance** = 300 m - **True length of chain** = 30.1 m - **Nominal length of chain** = 30 m So, \[ \text{Actual distance} = 300 \times \frac{30}{30.1} \] \[ = 300 \times .99667774 \approx 299.003 \text{ m} \] --- # Final Answer \[ \boxed{299. \text{ m}} \] **The actual distance is approximately 299. m.**